Abstract

We address some forward and inverse problems involving indefinite eigenvalues for discrete -Laplacian operators with potential terms. These indefinite eigenvalues are the discrete analogues of -Laplacians on Riemannian manifolds with potential terms. We first define and discuss some fundamental properties of the indefinite eigenvalue problems for discrete -Laplacian operators with potential terms with respect to some given weight functions. We then discuss resonance problems, anti-minimum principles, and inverse conductivity problems for the discrete -Laplacian operators with potential terms involving the smallest indefinite eigenvalues.

1. Introduction

In this paper, we study a generalized version of spectral theory, resonance problems, antiminimum principles, and inverse problems for discrete -Laplacian operators with potential terms on a network. We define a network as a way of interconnecting any pair of users or nodes by means of some meaningful links. Therefore, we represent a network by a weighted graph with a weight function.

The main goal of this paper is to characterize the indefinite eigenvalues and to solve the inverse conductivity problems for the equations
where and are real valued functions on a network with boundary . Here, is the discrete -Laplacian on a network with weight defined by
for . To address these problems, many researchers have especially concentrated on spectral graph theory which has been one of the most significant tools used in studying graphs. This has led to noteworthy progress in the study of these questions (see, e.g., [1, 2]). In this paper, we are primarily concerned with indefinite eigenvalue problems.

In particular, we deal with these problems under the assumptions that is positive and that has both positive and negative values. For each case, we present properties for the smallest indefinite eigenvalue as follows:(i)the variationally expressed form of ,(ii)the positivity of eigenfunctions corresponding to ,(iii)the multiplicity of .

Moreover, we also show that is isolated. Using these properties, we then discuss resonance problems, antiminimum principles, and the inverse conductivity problems. Note that the uniqueness of the conductivity is not guaranteed from . This implies that there can be different conductivities and on edges such that the smallest indefinite eigenvalues of networks for and are the same. Therefore, to guarantee the uniqueness of the conductivity, we impose the additional constraint, the monotonicity condition, on conductivity of the edges. The result for the case that is positive is Theorem 10 and the results for the other case of are Theorems 18 and 19.

Recently, in order to expand the results on spectral graph theory with respect to the above viewpoint, great efforts have been concentrated on studying the properties of graphs involving eigenvalues of operators such as discrete Schrӧdinger or discrete -Laplacian operators (see, e.g., [3–8]) which are generalizations of the discrete Laplacian. In [9], in particular, Amghibech introduces the indefinite eigenvalue problem for the case where and in (1) and gives some characterizations of the smallest indefinite eigenvalue. The author also addresses a resonance problem, an antiminimum principle, and an inverse problem.

This paper is organized as follows. In Section 2, we recall some basic terminology and properties of networks. In Section 3, for the case that is positive, we give some characterizations of the smallest positive indefinite eigenvalue, and we study the resonance problems, the antiminimum principles, and the inverse conductivity problems. Finally, in Section 4, we discuss the same problems discussed in Section 3 under the assumption that has both positive and negative values.

2. Preliminaries

In this section, we describe the theoretic graph notations frequently used throughout this paper.

By a graph we refer to a finite set of vertices with a set of two-element subsets of whose elements are called edges.

For notational convenience, we denote by the fact that is a vertex in . A graph is said to be a subgraph of if and . If consists of all the edges from which connect the vertices of in , then is called an induced subgraph. Throughout this paper, we assume that the graph is finite, simple, and connected.

A weight on a graph is a function satisfying(i),
(ii) if ,(iii) if and only if ,

and a graph with a weight is called a network . The integration of a function is defined by

For an induced subgraph of , by we denote a graph whose vertices and edges are in and vertices in for some . Here, and are called interiors and boundaries, respectively.

The -gradient of a function is defined as
for . In the case of , we write simply instead of .

It has been known that for any pair of functions and , we have
where for and . This fact yields many useful formulas such as the network version of the Green theorem (for details, see [7]).

For the given functions and , if and satisfy (1), then is called the (Dirichlet) indefinite eigenvalue for where and is called an eigenfunction corresponding to . Moreover, is called an indefinite eigenpair.

Finally, we recall some known results on discrete -Laplacian operators such as the minimum principle and Picone's identity.

Theorem 1 (see [6] minimum principle for on networks). Let satisfy the differential inequality for all . If attains the minimum at a point in , then is constant in .

Theorem 2 (see [9] Picone's identity for on networks). For an induced subnetwork of a given weighted network , let two functions and be nonnegative and positive on , respectively. Then
for all in . Moreover, if the induced subnetwork is connected, then the equality holds if and only if there exists such that for all in .

3. Indefinite Eigenvalue Problems with Positive Weight Functions

In [9], Amghibech introduces the indefinite eigenvalue problems for on networks with standard weights. In this paper, we study the indefinite eigenvalue problems under more complicated situations than those of Amghibech. More specifically, we look at the -Laplacian operator combined with potential terms and moreover, we do not impose any restrictions on the weight of the networks, further differentiating this paper from [9].

We now start this section under the assumption that is positive.

3.1. The Smallest Indefinite Eigenvalue

In this subsection, we prove the existence of the smallest indefinite eigenvalue for when is positive. We also address some fundamental problems such as the multiplicity of and its isolation.

It will be shown in the next theorem that exists and can be variationally expressed as
where

Theorem 3. There exists a nonzero function such that
Moreover, is the smallest eigenvalue for and is an eigenfunction corresponding to .

Proof. Note that
where . Here, we note that is closed and bounded (i.e., compact), since it is a subset of vectors in , for , and since is positive. Therefore, there exists such that
Since it is easily seen from (1) and (5) that for each eigenvalues , it suffices to show that is an eigenpair. For any , we define a function as
Taking an arbitrary , we have
for a sufficiently small and
Hence, we have
for a sufficiently small . Note that the right-hand side is continuously differentiable with respect to and equals zero at . Thus, we have
Since is chosen arbitrary in , we have
which completes the proof.

We now prove the simplicity of . To achieve this goal, we first prove a theorem which asserts that there always exists an eigenfunction corresponding to which is positive in .

Theorem 4. There exists with in such that is an indefinite eigenpair for .

Proof. It follows from Theorem 3 that there exists an eigenfunction corresponding to satisfying
Let . Then
Thus, we have
Otherwise, by the definition of ,
Thus,
It follows from Theorem 3 that is an indefinite eigenpair. Now it suffices to show that in . Suppose, to the contrary, that for some . It will be shown that . Since is an eigenvalue, it follows from (1) that
and thus for all where means that two vertices and are connected by an edge. By repeating the above process for , we conclude that for each . Since the network is assumed to be connected, for all .

Using the above theorem, we prove the simplicity of as follows.

Theorem 5. If is an indefinite eigenpair for , then

Proof. As shown in the proof for Theorem 4, if is indefinite, then is also an indefinite eigenpair. Let for all in . Then we have
which implies that
Since
for all in , we have
for all , in . Hence, either or for all in .

The above theorem shows that the dimension of the eigenspace corresponding to is one. Thus, we have the following.

Corollary 6. The multiplicity of is one.

For linear operators such as on finite networks, it is clear that the number of eigenvalues (including multiplicity) is the same as the number of vertices. However, when we consider nonlinear operators such as , it becomes significantly more complicated to count the number of eigenvalues. It is not sufficient to simply prove whether the number of eigenvalues is finite of infinite. However, by applying Picone's identity, it is possible to show that the smallest indefinite eigenvalue is isolated for a set of indefinite eigenvalues.

Theorem 7. The smallest eigenvalue is isolated.

Proof. We proceed by contradiction. Suppose that for each , there exists satisfying and
Since the multiplicity of is one, there exists an eigenfunction corresponding to with in such that in as . Hence, for sufficiently small we have in . Since
we have
That is,
Multiplying and integrating over on both sides (32) and using Picone's identity, we have a contradiction.

3.2. Resonance Problems, Antiminimum Principle, and Inverse Problems

In this subsection, we deal with some interesting problems such as the resonance problems, the antiminimum principles, and the inverse conductivity problems with regard to indefinite eigenvalues. We remind the reader that during this section, we assume the weight function is a positive valued function.

For a given function and a nonnegative source term , we consider the following equation:
It is clear that the above equation has a solution (in fact, an eigenvalue) if in . The next result shows that if , then the converse of the statement also holds. Thus, there is no solution of the above equation if is nonzero in .

Theorem 8 (resonance problem). Suppose that a function satisfies the condition that the smallest indefinite eigenvalue is positive. Then (33) has a solution if and only if .

Proof. Suppose that a function is a solution to the equation and we define a function as
Since it is obvious that if , then ; we assume that . Then we have
which implies that for some . If then is an eigenfunction corresponding to so that . Now suppose that so . Since and , we have
Thus, by using a similar method that we used in the proof for Theorem 4, it is easy to we show that the solution is positive in . Using Picone's identity, we have
which implies that .

The next theorem is the antiminimum principle. From it, we see that each (nonconstant) solution for the following equation
has its minimum in if .

Theorem 9 (antiminimum principle). For a nonnegative source term , suppose is a solution to the following equation:
If , then for some .

Proof. By virtue of Theorem 8, it suffices to show that if there exist a nonnegative solution for (75), then . Suppose is a solution to (75) with , . Using a similar method that we used in the proof of Theorem 4, we can easily show that if for some , then . Thus, we may assume that is positive in . By Picone’s identity, we have
where is the positive eigenfunction corresponding to . Thus, we have
Since , we finally have , which completes the proof.

We now discuss an inverse conductivity problem on networks. The main concern is related to the problem of recovering the conductivity (weight) of the network by the smallest indefinite eigenvalue for with respect to . Note that the uniqueness of the conductivity is not guaranteed by . This implies that there can be different conductivities and on the edges which induces the same eigenvalue for the operators . To guarantee the uniqueness of the conductivity, we need to impose some more assumption on the structure of network or on the conductivity. We impose here the additional constraint, called the monotonicity condition, on the conductivity of the edges. The main result of this section shows that there are no different conductivities and on the edges satisfying in which induce the same smallest indefinite eigenvalue .

Theorem 10 (inverse conductivity problem). For networks for , let be the smallest indefinite eigenvalue for . If the weight functions satisfy
then one has
Moreover, if and only if one has(i) on ,(ii) whenever or where is the eigenfunction corresponding to , .

Proof. By definition of the smallest eigenvalue, we have
It follows from that
Hence we have . Now, we suppose that . Then
Since , we have
Thus whenever , which implies that
Thus . Hence whenever , . If whenever , , then
Thus we have .

4. Indefinite Eigenvalue Problems with Weight Functions Which Have Both Positive and Negative Values

In this section, we address problems for the other case that has both positive and negative values. Namely, we now assume that the function satisfies
where
for .

4.1. Indefinite Eigenvalue Problems

We now discuss the indefinite eigenvalue problems with the assumption that has both positive and negative values and two real values and defined by

Theorem 11. If functions and satisfy either or in , then there exists such that
where
Moreover is the smallest positive eigenvalue for and is an eigenfunction corresponding to .

Proof. Define
We note that is not compact and its boundary is given by
so is compact. Now we take such that converges at some point . Since converges, also converges. From and the function which satisfies either or , we easily show that
Therefore, there exists such that
Now we take an arbitrary . Since for sufficiently small , by definition of , we have
Thus,
for a sufficiently small . The right-hand side is continuously differentiable with respect to and is equal to zero at . Thus,
Since the above equations hold for an arbitrary , we have

Theorem 12. For a function and satisfying either or in , there exists such that
where
Moreover is the largest negative eigenvalue for and is an eigenfunction corresponding to .

Proof. Since the proof is similar to that of the previous theorem, we omit it.

We note that it follows from the two above results that if either or , then there exist and at the same time. The specific case of was dealt with in [9].

In the following results, we give some properties of and its eigenfunction. One also can get similar results for and its eigenfunction, assuming that the function satisfies either or .

Theorem 13. For a function and a weight function satisfying either or in , there exists a positive eigenfunction corresponding to the indefinite eigenvalue for .

Proof. It follows from Theorem 11 that there exists an indefinite eigenfunction satisfying
for . Let for all in . Then and
Thus, we have
Otherwise, by definition of ,
Thus,
It follows from Theorem 11 that is an indefinite Dirichlet eigenpair. Next we show that for all in . It is sufficient to prove that if there exists in such that , then . Since is an indefinite Dirichlet eigenpair, it satisfies (1). This implies that
Hence for all . By repeating the above process for , we conclude that for each . Since is a connected network, for all . However, this contradicts the fact that . Thus, for all in .

Corollary 14. For a function with in or in , if is an eigenfunction corresponding to the eigenvalue for with respect to , then for all , in .

Proof. Let be an eigenfunction corresponding to . Then by Theorem 13, is also an eigenfunction corresponding to . Therefore, we have
This implies that for all . Thus, we have , .

Corollary 15. The multiplicity of is one.

4.2. Resonance Problems, Antiminimum Principle, and Inverse Problems

As previously mentioned, throughout this section the weight function is assumed to have both positive and negative values in . The next theorem shows that even in this case, we can solve a resonance problem similar to that in Theorem 8.

Theorem 16 (resonance problems). Suppose that a function satisfies . For , the equation
has a solution if and only if . Moreover, the solutions are eigenfunctions corresponding to .

Proof. Suppose that a function is a solution to (72). If , then we have . Suppose and set a function as for all . Since for all , for all . Since is a solution of (72), we have
which implies that for some . Assume ; then is an eigenfunction corresponding to so that . Now, assume . Then . Suppose for some . Then we have . Since is a nonnegative function, we have for , so . This presents a contradiction. Thus, , . Let be a positive eigenfunction corresponding to . By Picone’s identity,
which implies that .

The next result that we will discuss is the parallel version of the antiminimum principle discussed in Theorem 9 where the weight function is assumed to have both positive and negative values in .

Theorem 17 (antiminimum principle). Let a function and a weight function with in or in be given. For a nonnegative source term , suppose is a solution to the following equation:
If , then for some .

Proof. By virtue of Theorem 16, it suffices to show that if there exists a nonnegative solution of (75) then . Suppose for some . Then we have . Since is a nonnegative function, we have for , so . This is a contradiction to the assumption. Thus, we have in . Let be a positive eigenfunction corresponding to . By Picone's identity,
Since , we have

Finally, we deal with inverse conductivity problems for and .

Theorem 18 (inverse conductivity problem). For networks , , let be the smallest positive indefinite eigenvalue for . One can suppose that the given functions and satisfy either or . If the weight functions satisfy
then one has
Moreover, if and only if one has(i) on ,(ii) whenever or ,where is the eigenfunction corresponding to , .

Proof. Let be an eigenfunction corresponding to for . By the definition of the smallest eigenvalue, we have
It follows from that
Thus, we obtain . Now, we suppose that . Then
Since , we have , . Then whenever . This implies that
Hence, is an eigenfunction of . Since is simple, we have . Therefore, whenever , .If whenever , , then
Thus, we have .

Theorem 19 (inverse conductivity problem). For networks , , let be the largest negative indefinite eigenvalue for . One can suppose that the given functions and satisfy either or . If the weight functions satisfy
then one has
Moreover, if and only if one has(i) on ,(ii) whenever or ,where is the eigenfunction corresponding to , .

Proof. The proof is similar to that in Theorem 18 and we thus omit it.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MOE) (no. 2012R1A1A2004689).